perms.swap

Swap is used to apply the swap function within a quantum state or an operator.

Functions

swap(rho[, sys, dim, row_only])

Swap two subsystems within a state or operator.

Module Contents

perms.swap.swap(rho, sys=None, dim=None, row_only=False)

Swap two subsystems within a state or operator.

Swaps the two subsystems of the vector or matrix rho, where the dimensions of the (possibly more than 2) subsystems are given by dim and the indices of the two subsystems to be swapped are specified in the 1-by-2 vector sys.

If rho is non-square and not a vector, different row and column dimensions can be specified by putting the row dimensions in the first row of dim and the column dimensions in the second row of dim.

If row_only is set to True, then only the rows of rho are swapped, but not the columns – this is equivalent to multiplying rho on the left by the corresponding swap operator, but not on the right.

Examples

Consider the following matrix

\[\begin{split}X = \begin{pmatrix} 1 & 5 & 9 & 13 \\ 2 & 6 & 10 & 14 \\ 3 & 7 & 11 & 15 \\ 4 & 8 & 12 & 16 \end{pmatrix}.\end{split}\]

If we apply the swap function provided by toqito on \(X\), we should obtain the following matrix

\[\begin{split}\text{Swap}(X) = \begin{pmatrix} 1 & 9 & 5 & 13 \\ 3 & 11 & 7 & 15 \\ 2 & 10 & 6 & 14 \\ 4 & 12 & 8 & 16 \end{pmatrix}.\end{split}\]

This can be observed by the following example in toqito.

>>> from toqito.perms import swap
>>> import numpy as np
>>> test_mat = np.array([[1, 5, 9, 13], [2, 6, 10, 14], [3, 7, 11, 15], [4, 8, 12, 16]])
>>> swap(test_mat)
array([[ 1,  9,  5, 13],
       [ 3, 11,  7, 15],
       [ 2, 10,  6, 14],
       [ 4, 12,  8, 16]])

It is also possible to use the sys and dim arguments, it is possible to specify the system and dimension on which to apply the swap operator. For instance for sys = [1 ,2] and dim = 2 we have that

\[\begin{split}\text{Swap}(X)_{2, [1, 2]} = \begin{pmatrix} 1 & 9 & 5 & 13 \\ 3 & 11 & 7 & 15 \\ 2 & 10 & 6 & 14 \\ 4 & 12 & 8 & 16 \end{pmatrix}.\end{split}\]

Using toqito we can see this gives the proper result.

>>> from toqito.perms import swap
>>> import numpy as np
>>> test_mat = np.array(
...     [[1, 5, 9, 13], [2, 6, 10, 14], [3, 7, 11, 15], [4, 8, 12, 16]]
... )
>>> swap(test_mat, [1, 2], 2)
array([[ 1,  9,  5, 13],
       [ 3, 11,  7, 15],
       [ 2, 10,  6, 14],
       [ 4, 12,  8, 16]])

It is also possible to perform the swap function on vectors in addition to matrices.

>>> from toqito.perms import swap
>>> import numpy as np
>>> test_vec = np.array([1, 2, 3, 4])
>>> swap(test_vec)
array([1, 3, 2, 4])
Raises:

ValueError – If dimension does not match the number of subsystems.

Parameters:

  • rho (numpy.ndarray) – A vector or matrix to have its subsystems swapped.

  • sys (list[int]) – Default: [1, 2]

  • dim (list[int] | list[list[int], int, numpy.ndarray]) – Default: [sqrt(len(X), sqrt(len(X)))]

  • row_only (bool) – Default: False

Returns:

The swapped matrix.

Return type:

numpy.ndarray